**This article was originally published 6 March 2020.**

**This article was originally published 6 March 2020.**

#### BDO Australia’s Martin Emilson discusses selecting the appropriate risk-free rate based on cash-flow duration.

### Introduction

In valuation theory, when determining the risk-free rate, the valuer should match the risk-free security with the period in which the cash flows to be valued are expected.

It is common market practice to use a 10-year or 20-year government bond yield in the discount rate build-up when valuing a going concern with perpetual life. When valuing contracts (or 'right-to-use' assets/obligations in the case with the new IFRS 16 lease standards), cash flows are expected over a discrete period (e.g. 10 years) and it is common for the practitioner to match the contract length with a risk-free security of similar length (e.g. a 10-year government bond yield when valuing a 10-year contract).

In this article, which I hope will be useful to valuation practitioners and auditors, I provide examples of how the application of 10-year or 20-year risk-free yields when valuing a going concern business is often a correct assumption. However, the application of a 10-year yield when valuing a contract running for 10 years is almost always incorrect. The latter observation is of extra importance given the introduction of the new lease standards (IFRS 16) where valuation practitioners are asked to provide (or review) the incremental borrowing rate (IBR) for a lessee.

### The concept of duration

The matching of the risk-free security with the period in which the cash flows to be valued are expected is closely connected with the concept of duration. Duration has its roots in risk management and is a measure of interest-risk sensitivity of assets (and liabilities). Duration, which is the weighted average of times until the underlying cash flows are received, helps tell investors: i) how long it will take on average to get their original investment back; ii) how sensitive an asset (e.g. a bond) is to changes in interest rates. Theoretically, the longer an asset's duration, the more sensitive it is to changes in interest rates. To manage interest rate risk, strategies of duration-matching and immunisation are commonly applied and involve matching the duration of the liability with the duration of the asset.

Similarly, duration-matching is important when discounting a set of cash flows to present value (PV). In theory, it is correct to discount each year's expected net cash flow estimate using a matched maturity risk-free rate. That is, cash flow in year one to be discounted by a one-year government bond rate plus asset risk^{1}, cash flow in year two to be discounted by a two-year government bond rate plus asset risk… cash flow in year 15 to be discounted by a 15-year government bond plus asset risk, and so on. In practice, and as a practical compromise, when valuing long-term investments and going-concern businesses, valuers generally use a simplifying assumption of long-term government bonds as the risk-free security when discounting all years' cash flows. The time to maturity of the long-term government bond serves as a proxy of the weighted duration of the cash flows of the asset being valued, with some cash flows expected before the term of the government bond while other cash flows are expected thereafter.

### Illustrative examples – 1. Duration of cash flows for a going-concern business

It is sometimes argued that when valuing a going concern, the valuer should match the cash flows with a discount rate based on a risk-free security with the longest available maturity (30 years or longer). In the below illustrative example, based on the discounted cash flow (DCF) methodology, I show that the duration of cash flows for a going-concern business is actually much shorter in most instances.

In this example, the DCF generates a value of $100 when assuming equity cash flows of $10, perpetual equity cash flow growth of 2%, and cost of equity (CoE) of 12% (see Table 1 below).

To calculate the duration, I have used the Macaulay duration formula (which weight the cash flows based on market-value/present value).^{2} As shown in Table 1 below, the duration is 11.2 years, based on the above cash flows ($10 growing at 2% into perpetuity), a CoE of 12%, and end-year discounting.^{3}

**Table 1 – Illustrative example of cash flow duration**

Apart from the timing of cash flows, duration is a function of the equity discount rate and growth in equity cash flows. As duration is the result of an infinite combination of discount rates and growth rates, Table 2 illustrates the sensitivity of cash flow duration when altering these. I have also cross-checked my results to some of the workings published by Shannon Pratt, a well-known authority in the valuation community, and note that results are consistent.^{4}

**Table 2 – Duration sensitivity (years)**

Table 2 highlights some interesting takeaways. It shows us that applying a risk-free security with a maturity of 30 years would be appropriate only when valuing going-concern businesses with relatively low risk (CoE of 8-9%) and high perpetual growth (4-5%). Instead, it suggests much shorter duration for the stock market as a whole (large cap). Today's expected equity market returns in developed financial markets of approximately 7%-10%^{5} in conjunction with modest expectations of perpetual growth (1.5%-2.5%) suggest durations in years from lower teens to approximately 20. Furthermore, when valuing smaller businesses (relative large-cap companies) durations of approximately 10 years do not appear unreasonable given their higher cost of equity (small-cap premiums, etc.).^{6}

Now, having established that the duration of equity cash flows growing at 2% into perpetuity is 11.2 years (assuming CoE of 12%) we can put the results to the test by using an immunisation strategy. Let us assume that an investor invests $100 in a company with equity cash flows of $10, perpetual equity cash flow growth of 2%, and cost of equity (CoE) of 12%. The investor is financing the investment by selling $100 worth of a risk-free security (short-selling). At the same time, the investor wants to make sure the interest rate risk is limited and decides that selling $100 worth of a 2% yield, zero-coupon bond maturing in 11 years will do the trick (consistent with duration matching).

Figure 1 shows the development in value of each instrument (bond and shares) given shifts in market interest rates following the investment.^{7} With duration matching and immunisation, the movement in values should be equal.^{8} As can be seen in Figure 1, a downward parallel shift in market interest rates (yield curve) of 1% increases the value of the bond and shares from $100 to $111. An upward shift of 1% decreases the value of the bond and shares to $90 and $91, respectively.

**Figure 1 – Development in values following shifts in interest rates – Shares vs bond (11-year term)**

Now, let us assume the investor thinks that because the equity cash flows will be received in perpetuity, a better strategy is to match the equity investment with a bond with a longer maturity and decides to go with a 30-year government bond. As previously mentioned, when comparing two instruments, the one with the longest duration will be more sensitive to changes in interest rates and this is exactly the results in Figure 2.

**Figure 2 – Development in values following shifts in interest rates – shares vs bond (30-year term)**

To sum up, matching the equity cash flow duration with a longer-term bond (in this case a 30-year government bond) would not be consistent in this case. In fact, data presented in the illustrative example above would suggest durations in years from low teens to approximately 20 and closer to 10 years for smaller companies.

### Illustrative examples – 2. Duration of cash flows for a lease

The next area is the cash-flow duration of contracts with finite lives (e.g. leases). As previously mentioned, it is common for valuation practitioners to match the contract length with a risk-free security of similar length (e.g. a 10-year government bond yield when valuing a 10-year contract). Applying the same concept of duration, the below illustrative example shows that such assumption is theoretically incorrect.

In this example, let us assume a company (lessee) has entered into a lease that gives the lessee the right to use the underlying asset for 10 years. In return, the lessee is paying annual rent (at the end of each year) of $10 inflated by 3.5% a year (nominal contract value of $117.30). To discount the lease payment to present value, the valuer has to determine the lessee's incremental borrowing rate (IBR).^{9} In this example (see Table 3), I have assumed that the IBR is made up of Australian government bond yields as of 31 December 2019 plus an additional and constant asset-specific risk premium of 4%.^{10} As a reference, I have also included IBRs based on US government bond yields as of the same date and note that both were upward sloping.

**Table 3 – IBR assumptions**

If the valuation of the lease contract is based on a discount rate of 5.38% (AUS IBR 10-year tenure), the calculated value is $87.70 (see Table 4). However, the value is $88.50 if applying matching discount rates to each year's lease payments (see Table 5).^{11} It should be mentioned that the yield curve at the end of 2019 reflected fears of a slowdown in the economy and that the difference in value would have been higher with the steeper yield curves of the past.

**Table 4 – Value of lease assuming a 10-year discount rate (IBR)**

**Table 5 - Value of lease assuming maturity matching**

Again, let us put the results to the test by using an immunisation strategy using the same methodology as in the first illustrative example. As the cash-flow duration of the above 10-year lease is approximately 5 years, the alternative investment is based on a 5-year risk-free security. As can be seen in Figure 3, the interest rate risk is almost eliminated with this approach. However, the assumption of a 10-year IBR when discounting the lease payments does not result in immunisation (see Figure 4).

**Figure 3 – Development in values following shifts in interest rates – lease value vs bond (5-year term)**

**Figure 4 – Development in values following shifts in interest rates – lease value vs bond (10-year term)**

Now, the theoretically correct approach to discount each lease payment with a matching discount rate is cumbersome, especially if the lease contract stipulates monthly payments as opposed to yearly.^{12} A simplifying assumption would then be to use one weighted discount rate for all lease payments (and for bundles of leases with similar characteristics). This weighted discount rate ideally would be one that generates the same value as with the detailed approach (similar to the internal rate of return concept). One would assume that you would find a perfectly correct rate by deriving the duration and by selecting a rate that is consistent with that duration. Well, the bad news is that due to the yield curve not being flat, the rate selected (which is a market-value-weighted rate-based on the Macaulay duration formula) may not necessarily match the IRR. Typically, it is an underestimate, that is, the market-value-weighted rate is lower than the IRR, the more so the steeper the yield curve. In this example, and guided by the numbers in Table 3, the duration of 5.3 years would indicate a market-value-weighted rate of about 5.10%, which is slightly less than the IRR of 5.19% (consequently, the lower rate overvalues the lease slightly to $88.90). It turns out, the only way to get the rates to exactly match is through the application of some technical mathematics, including the concept of dollar duration weights (basis point weights) as opposed to market-value weights.^{13}

To sum up, the application of a rate-based on dollar duration weights would be optimal. However, in most instances a value-weighted rate (or even time-weighted rate) is certainly more correct than applying rates based on the total term of the lease and the more so the steeper the yield curve.

### Conclusion

The examples in this article show that the application of 10-year or 20-year risk-free yields when valuing a going-concern business is often a correct assumption. However, apply a 10-year yield when valuing a contract with regular payments running for 10 years is almost always incorrect (only correct if the underlying yield curve is flat).

*Martin Emilson is a Director of BDO's Corporate Finance practice in Melbourne, Australia. He can be reached on +61 4 36 839 814 and martin.emilson@bdo.com.au*

**References**

*Cost of Capital: Applications and Examples*, Shannon P. Pratt and J.Grabowski, Fourth Edition, 2010, page 119.

Equity market risk premiums, Professor Aswath Damodaran (http://people.stern.nyu.edu/adamodar/pc/datasets/ctryprem.xls).

*Bond Math: The Theory Behind the Formulas*, Donald J. Smith, 2011.

IFRS 16 Leases

S&P Capital IQ

**Notes**

1. In CAPM represented as the beta factor times the equity market risk premium.

2. Frederick Macaulay originally introduced the formula to measure the duration of bonds but the formula can easily be used to calculate the duration of any set of cash flows. (https://en.wikipedia.org/wiki/Bond_duration)

3. Assuming that all the equity cash flows are distributed as dividends at the end of each year.

4. Shannon Pratt shows that the calculated duration of cash flows is 10.5 years when assuming a discount rate of 15% and perpetual growth of 4%. Cost of Capital: Applications and Examples, Shannon P. Pratt and J. Grabowski, Fourth Edition, 2010, page 119.

5. Based on equity market risk premiums published by Professor Aswath Damodaran (http://people.stern.nyu.edu/adamodar/pc/datasets/ctryprem.xls).

6. It should be mentioned that there are other factors to consider when selecting the appropriate risk-free rate. The selection of the risk-free rate should be consistent with the risk-free rate used when determining the equity market risk premium (the equity market risk premium is the difference between the expected return on a market portfolio and the risk-free rate). Furthermore, the size and liquidity of the market for government bonds should be considered.

7. Assuming a parallel shift in the market yield curve and that all other assumptions (e.g. equity cash flows and growth) remain the same.

8. Duration matching and immunisation only works perfectly when rates move in small increments. Duration is a linear measure of how the price of an asset changes in response to interest rate changes. As interest rates change, the price does not change linearly, but rather is a convex function of interest rates.

9. Defined in IFRS 16 Leases as 'The rate of interest that a lessee would have to pay to borrow over a similar term, and with a similar security, the funds necessary to obtain an asset of a similar value to the right-of-use asset in a similar economic environment.'

10. Determining the asset risk premium (or the financing spread adjustment) is not part of the scope of this article.

11. The only time that the results from Table 4 and 5 will match is when the yield curve is flat. Usually, both the risk-free rate and financing spread are increasing with the term of the instrument.

12. Monthly payments over ten years would require 120 individual discount rates.

13. I have not elaborated further on this but the interested reader can deep-dive into the concept of dollar duration in the book "Bond Math: The Theory Behind the Formulas", by Donald J. Smith (2011, John Wiley & Sons).